Last update: 18.03.2005, 13:00 CET

PROOF OF THE FIRST CARTAN EQUATION USING THE TETRAD POSTULATE

with a Commentary by G.W. Bruhn and W.A. Rodrigues Jr.

The first Cartan equation is

Ta = d Ù qa + ωab Ù qb                                                             (1)

i.e.

Taμν = ∂μqνa − ∂νqμa + ωμab qνbωνab qμb                                         (2)

Here Ta is the torsion two-form; qμb is the tetrad one-form; ωμab is the spin connection.

The torsion tensor is:

Tλμν = qaλ Taμν .                                                             (3)

Use the tetrad postulate in the form

Γλμν = qaλ (∂μqνa + ωμab qνb)                                                             (4)

>From eqns (2) - (4):

Tλμν = qaλ (∂μqνa + ωμab qνb) − qaλ (∂νqμa + ωνab qμb)

Tλμν = Γλμν − Γλνμ                                                             (5)

Equ.(5) is the torsion tensor of Riemann geometry. Given the tetrad postulate (4) it has been proven to be the Cartan equation (1). QED

Commentary

The proof should start with Equ.(5), with the definition of the torsion tensor

Tλμν = Γλμν − Γλνμ .                                                             (5)

Using the compatibility relation of frames cf. [2; (3.9-10)] (not the tetrad postulate as is remarked by Evans) in the form

qλa Γλμν = ∂μqνa + qνb ωμab                                                   (4')

and the relation

Taμν = qλa Tλμν .                                                             (3')

we obtain from (5)

Taμν = (∂μqνa + qνb ωμab) − (∂μqνa + qνb ωμab) .                                         (2')

Remark 1
The expressions in brackets are the type (II) covariant derivatives of qνa (cf. [1; (3.2)])

D(II)μqνa = ∂μqνa + qνb ωμab

hence

Taμν = D(II)μ qνaD(II)ν qμa .                                                   (2'')

This can be rewritten with some easy algebra in form of an exterior derivative:

Ta = TaμνdxμÙdxν = D Ù qa := dÙqa + ωab Ù qb .                                         (1')

Remark 2
In contrast to Evans' denotation Ta is not the torsion two-form, but represents for a = 0, 1, 2, 3 four two-forms T0, T1, T2 and T3 .

Remark 3
We impose that the covariant derivative of the tetrad 1-form appearing in Equ. (1') is of type (II). Hence it is shown that different covariant derivative types of the tetrad tensor qνa can appear depending on the context.

References

[1]    G.W. Bruhn and W.A. Rodrigues Jr.: Covariant Derivatives of Tensor Components,